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Elementary school students engaging iElementary school students engaging in making generalisationn making generalisation

Presenters: Wei-Chih Hsu Presenters: Wei-Chih Hsu Professor : Ming-Puu ChenProfessor : Ming-Puu ChenDate : 08/19/2008Date : 08/19/2008

Yeap, B.H. & Kaur, B. (2008). Elementary school students engaging in making generalisation. ZDM: International Journal in Mathematics Education, 40, 55-64.

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OutlineOutline

• Introduction

• Literature review

• Methodology

• Discussion & Conclusion

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IntroductionIntroduction

• This article reports on – the generalisation strategies used by students in a gr

ade five class in an elementary school in Singapore.– the factors influence generalisation strategy use.

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Literature review (1/4)Literature review (1/4)

• Generalisation and school mathematicsGeneralisation and school mathematics– Algebraic concepts be introduced to students in elementary a

nd middle school years. (Kaput, 1995)

– Expressing generality as one of the roots of algebra. (Mason, Graham and Johnston-Wilder, 2005)

– Algebraic thinking ‘‘involves acts of deliberate generalisation and expression of generality. (Lins and Kaput, 2004)

– Generalisation is the heartbeat of mathematics. (Mason, 1996)

• The Singapore mathematics curriculum and the teachinThe Singapore mathematics curriculum and the teaching of algebrag of algebra– The Singapore mathematics curriculum does not include muc

h formal algebra in the elementary school.– Students are introduced to formal algebra only in grade six (a

ged 12) (Ministry of Education of Singapore, 2006).

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Literature review (2/4)Literature review (2/4)

• The Singapore mathematiThe Singapore mathematics curriculum and the teacs curriculum and the teaching of algebraching of algebra– Figure 1 shows two tasks f

ound in one of the textbooks (Collars, Koay,Lee, Ong & Tan, 2006).

• Tasks that require students to generalise and use algebraic expressions to describe general terms are not common.

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Literature review (3/4)Literature review (3/4)

• The Singapore mathematics cThe Singapore mathematics curriculum and the teaching of urriculum and the teaching of algebraalgebra – In an analysis of 190 recent r

eleased test items, only two items were found to include some kind of generalisation (Yeap, 2007).

• One of them (Fig. 3) required students to find a given term in a repeated pattern.

• The other item (Fig. 4) embedded a pattern into the context of a word problem.

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Literature review (4/4)Literature review (4/4)

• The generalisation strategies – Children used to come up with a four-category framework to

describe generalisation strategies. (Lannin, Barker & Townsend, 2006)

• (1) Recursive– describe a relationship that occurs in the situation between consec

utive values of the independent variable.

• (2) Chunking– build on a recursive pattern by building a unit onto known values

of the dependent variable.

• (3) Unitising– use a portion as a unit to construct a larger unit using multiples of

the unit.

• (4) Explicit– construct a rule that allows for immediate computation of the valu

e of the dependent variable for a given independent variable value.

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Methodology (1/2)Methodology (1/2)

• Participants– A grade five (aged 11 years) cl

ass from a typical school in Singapore was selected for the study.

– 38 students (although the students had differing abilities in problem solving, they had acquired the basic skills in arithmetic.)

• Instruments– Students were given a novel

task that comprises of several subtasks that required generalising.

– The selected task called Odd Numbers is shown in Fig. 6.

• It was not a typical textbook task.

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Methodology (2/2)Methodology (2/2)

• Instruments: The Odd Numbers task.– Required students to find a method for finding the sum of consecuti

ve odd numbers for a small number of terms.– Three types of subtasks were presented

• Recognise the given pattern of using relevant square numbers. • Developing a generalisation similar to the ones given in the example.

– near-transfer task. • Developing a generalisation that was less similar to the ones in the exa

mple.– far-transfer task.

• Procedure– Students were asked to describe what they did after they had compl

eted each subtask.– They were also asked a few additional questions in the post-task int

erview.– The data for each student was analysed at two levels

• (1) to identify the strategy used, • (2) to identify factors that facilitated the ability to generalize.

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Conclusions & Discussion (1/2)Conclusions & Discussion (1/2)

• The following factors were evidently important in the use of the generalisation strategies: – (1) the ability to see structures and relationships,

– (2) prior knowledge,

– (3) meta-cognitive strategies,

– (4) critical thinking strategies,

– (5) the use of organizing heuristics such as a table,

– (6) use of simplifying heuristics such as trying out simpler cases,

– (7) task familiarity,

– (8) technology.

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Conclusions & Discussion (2/2)Conclusions & Discussion (2/2)

• Lannin, Barker and Townsend (2006) have proposed three categories of factors that influence strategy selection in generalising: – (1) cognitive factors; – (2) task factors; – (3) social factors.

• In the present study, we focused on the cognitive factors and task factors as the students were observed individually.

• Future research should focus on – generalising strategies of mathematically able students and average,

or even lessable, – students to determine how the latter can reach the level of thinking

of mathematically able students. – A more comprehensive body of knowledge on

• how to make the ability to generalise accessible to all has important curricular and instructional implications and applications.